New and useful improvements in propergols or propellants

ABSTRACT

The present invention relates to propergols or propellants in the form of blocks which comprise two propergols or propellants having different speeds of combustion and, in accordance with the invention, the blocks have cross-sections according to which the inner contour of the propergol or propellant having the faster speed of combustion, has the shape of a star, and there is a separatrix between the two propergols or propellants also having the shape of a star, the number of branches of which is at least equal to the number of branches of the star shape forming the inner contour. The invention also relates to a method for calculating the shape of said blocks.

ite States Ptent Arribat Nov. 11, 1975 3,260,208 7/1966 Schluter .1 102/101 PII'HIHI') E.\'anzinerSamue1 Feinberg Attorney, Agent, or Firm-Brooks Haidt l-laffner & Delahunty [57] ABSTRACT The present invention relates to propergols or propellants in the form of blocks which comprise two propergols or propellants having different speeds of combustion and, in accordance with the invention, the blocks have cross-sections according to which the inner contour of the propergol or propellant having the faster speed of combustion, has the shape of a star, and there is a separatrix between the two propergols or propellants also having the shape of a star. the number of branches of which is at least equal to the number of branches of the star shape forming the inner contour. The invention also relates to a method for calculating the shape of said blocks.

13 Claims, 23 Drawing Figures U.S. Patent Nov. 11,1975 Sheet10f12 3,918,365

FIG. 2

US. Patent Nbv. 11, 1975 Sheet 2 of 12 3,918,365

US. Patent Nov. 11, 1975 Sheet4 of 12 3,918,365

US. Patent" Nov. 11, 1975 SheetS of 12 3,918,365

FIG. I!

FIG. 72

US. Patent Nov. 11, 1975 Sheet 9 of 12 3,918,365

FIG. I6

FIG. 77

US. Patent Nov.1l, 1975 Sheet 10 of 12 3,918,365

Patent Nov. 11, 1975 Sheet 11 of 12 3,918,365

US. Patent Nov. 11, 1975 Sheet 12 of 12 3,918,365

NEW AND USEFUL IMPROVEMENTS IN PROIPIERGOLS R PROPIELLANTS This is a continuation of copending US. patent application Ser. No. 181,909, filed Sept. 20, 1971 now abandoned.

The present invention relates to products of block form of specific shapes, made by assembling two materials called propergols (i.e. propellants) and capable of being converted into gas, more especially usable for the reaction propulsion of civil or military vehicles in space or in a gaseous or liquid medium. The invention also relates to a method for determining the characteristic surfaces of the propergols manufactured in accordance with the invention.

These blocks are intended for combustion in cham bers which are provided with one or more apertures, in which the gases produced during the combustion become pressurised with respect to the exterior, and which are generally cylindrical.

The outer lateral surface of a block in accordance with the invention has substantially the shape of a cylin der of revolution, and it is either adhered to the wall of the chamber, or inhibited, so as not to participate in the combustion. Along this surface and about its axis, the block has a central cavity, of elongated shape in the direction of this axis, set alight at the start of the firing; this cavity opens out at at least one of the ends of the block.

The block is manufactured with two separate homogeneous .propergols, intimately coupled one to the other with interruption along a surface of separation all in one piece, surrounding the central cavity.

The present invention is further illustrated and described by reference to the accompanying drawings, wherein FIG. I is a cross-section of a solid propergol prepared according to the invention;

FIG. 2 represents the relationships obtain in one sector of a typical propellant according to the invention;

FIGS. 3, land 5 illustrate the relationships between the shape functions of a propellant and the reduced thickness during various phases of combustion;

FIGS. 6, '7 and 8 show relationships in typical sectors of propellants according to the invention;

FIG. 9 shows relationships between the shape function and reduced'thickness of an embodiment during combustion;

FIG. Id shows a partial cross-section of initial conditions and flame fronts during combustion of an embodiment of the invention;

FIGS. i111 and 112 show relationships in sectors of propellants according to the invention;

FIGS. 13, I41 and 115 show families of curves representing flame fronts or surfaces of combustion in certain embodiments;

FIGSv I6, 117, I8, 19, 20, 21 and 22 illustrate the relationships in sectors of various embodiments of the invention; and

FIG. 23 shows relationships between the shape functions and radial distances in certain embodiments.

The propergols each burn, both separately and simultaneously, in substantially parallel layers, but their combustion speeds are different, and the quotient n of the rate of combustion of the rapid propergol divided by that of the slow propergol is substantially constant. The rapid propergol occupies the entire volume of the block between the central cavity and the surface of separation, and the slow propergol occupies the whole volume of the block between the surface of separation and the outer surface.

On a section of the block made in its cylindrical portion perpendicularly to the directions of the generatrices, and referring to FIG. 1 of the accompanying drawings, the following terms are applicable:

outer contour C is the outline of the outer surface it is constituted by a circle whose centre is marked 0 and whose radius has a length taken conventionally, in all that follows, equal to unity;

inner contour C is the outline of the inner surface; it is constituted by a closed curve, without double point, surrounding O, and within C separatrix G, is the outline of the surface of separation; it is constituted by a closed curve, Without double point, surrounding C within c,.., not touching either C or C flame front" C(r), is the outline of the surface formed by the points under ignition at a given instant I of the firing; at the instant of the ignition, C(t) is blended with C,,; at the instant when the combustion reaches the outer contour for the first time, C(t) is a curve C, touching C at at least one point; beyond C and as far as C inclusive, C(t) is a closed curve, initially within G, then cutting G, finally outside G; in the course of the firing, the first flame front touching G is marked C and the last one having one point at least in common with G is marked C The flame fronts or surfaces of combustion C C and C, are shown as dashed lines, and the drawing is hatched to indicate that the section represented by the sectional area between inner contour C and separatrix G comprises a first faster-burning propergol and the section at area between separatrix G and outer curve C,, represents a slower-burning propergol.

Each inner contour C has the general aspect of a star having p branches, with p being at least equal to 3; this star is more or less deformed, but it is still selected such that at the end of the rapid phase of the combustion (the phase when the rapid propergol alone burns), the flame front C is formed only of p consecutive arches which all turn their concavity towards the centre 0, which are each an element of curve parallel to a portion of the end of a diffeent branch of the star, and which are themselves substantially composed of arcs of circles whose centres, called main centres of curvature, are situated on p radii of C called main sides and each intersecting the inner contour at one point of the end of a different star branch.

The separatrix G has the general aspect of a star having 1/ branches, with v at least equal to p; this new star, more or less deformed, is still chosen and positioned in such a way that, on the one hand, at the start of the slow phase (the phase when the slow propergol alone burns), the flame front C itself also has the aspect of a star having 1 branches envelopping as it were the separatrix (FIG. I), and that, on the other hand, the flame front C, is formed only of 11 consecutive arches, turning their concavities towards 0, all being substantially tangential to C each being located opposite the end of a different branch of G, and being composed substantially of arcs of circles whose centres, called imagesummits", are allsituated in the vicinities of the ends of the 1/ branches of G.

Due to the shapes of the curves referred to above, the blocks in accordance with the invention are called bistellar blocks, whilst the conventional blocks having a single propergol and having an inner contour in the form of a star will be called, in contradistinction, monostellar blocks when reference is hereinafter made to them.

A first group of bistellar blocks is characterised in that the circle C, can be cut up fictitiously into a certain number p (at least equal to p) of sectors subtending equal angles at the'centre, and all containing portions of C substantially superimposable one on the other and portions of G substantially superimposable one on the other; these blocks have the advantage of an easier study of the successive flame fronts as from the mixed phase (when the two propergols burn together).

However, such a study is also facilitated on a second group of bistellar blocks, called symmetrical bistellars, characterised as follows by calling a sector of C,. limited by two consecutive main sides, the main sector; the portions of C and of G contained in any main sector each allow the inner bisectrix of the main sector to be the axis of symmetry; the main sides all intersect G at points situated on the ends of the star branches.

In this way, the evolution of the flame fronts in the blocksof the second group is effected without reciprocal influence of the different main sectors, and in each of these, it is also simplified by the existence of a symmetry in relation to the bisectrix of the main sector in question.

In a general manner, as from a main sector possessing such a symmetry, a sector of C,. limited by one of the sides and by the inner bisectrix of the main sector in question is referred to as an elementary sector. This bisectrix is the second side of the elementary sector, and the portions of C, and of G contained in an elementary sector constitute an elementary motif;

The study and the performance of a symmetrical bistellar block are deduced immediately, and in evident manner, from the properties of the main motifs", that is to say from the outlines of C and of G in the main sectors. However, these main motifs have properties which vary in a continuous 'manner with the value 2 Q for the angle at the centre of the main sector. In this way, it is possible in practice to limit the study to main sectors capable of producing blocks belonging to a third group, which is constituted by definition of the blocks of the second group having their main motifs all substantially superimposable one on the other.

In order to assist in considering the description of the invention, there follows a legend of symbols and abbreviations used in disclosing the invention and embodiments thereof:

LEGEND OF SYMBOLS n Combustion quotient, the ratio of rate of combustion in rapid propergol rate of combustion in slow propergol C Outer contour. lts radius is taken as l.

O Center of outer contour C C Outline of inner surface G Separatrix or surface of separation between the two propergols Does not touch C or C C(I) Curve of flame front surface at time 1 C Curve of first contact of flame front with C touches C at at least one point p Number of branches of star fonning C,,. Is a 3 C Curve of first flame front touching G v Number of branches of star forming separatrix G O Angle at center of main sector 1r/p Shape A Shape of C0 composed of circular arcs and straight line portions Shape N An inner contour of Shape A Shape (or Form) D composed of straight-line segments Final form of G e Radius of arcs constituting roundoffs at G segments Shape E= First outline of G c Class (v/ p) which is 2 l and at most =4 Cg Last flame front touching G ['(t) Perimeter of C(t) in rapid propergol 1"(!) Perimeter of C(t) in slow propergol u'(t) Flame radius in rapid propergol u"(t) Flame radius in slow propergol 11(1) u'(t) n u"(t) corrected distance of C(t) [(t) nl(t) I"(!) corrected perimeter of C(t) n =Selected coefficient chosen so [(1) is C( t) if situated solely in slow propergol Generally n n p=Filling coefficient =surface between C and C /inner surface of Ce; preferably p 0.9 o- Ratio of theroretical residual surface between C I and C /surface between C,, and C,,; At most 0.05 d) Shape function corrected perimeter of C(z)/corrected perimeter of C;

v Reduced thickness corrected distance of 4),, Shape function of initial flame front, i.e., on C d) First maximum value 4),. Minimum value in fast propergol (b Terminal value in fast propergol gb .=Values at C,,-- C in mixed phase, i.e., both fast and slow propergols d) Median value 4),, Value at C,,, first of final median phase (11 Final minimum value (b Value at C O ly,= Residual reduced thickness b Common value of distances to 0 from main centers of curvatures a Arithmetic means of distances to 0 from apeximages r k Shortest distance from G to a main center of curvature a, Greatest distance of G to O minimal shaping r o Radius of curvature of outer apices of C,

r,, Minimal value of r,, 0.06/bp e Radius of curvature at points of G situated in vicinity of outer apex Parameters of construction p, 0, r 6, b, a, r

Parameters of utilization r e, a p, 0', 4), (various values) Shape A of C Shape N =Normalised inner contour '10,, Angle of neutrality relative to m j Number of apex-images and geometrical centers of Shape D or E P Main center of curvature of sector a 1 Distance of ith apex-imagine from O B,- Inner apices of G r Radius of arcs of circles comprising C 11 Plane normal to axis K Maximum inherent locking m Number of different contours Z Lengths in direction of gas flow s Elementary inner perimeter Minimum value of shortest distance to main side.

Such blocks have, moreover, industrial interest, since it is generally not best for the main motifs to be different. However, if these motifs are all equal and each have their axis of symmetry, it is not absolutely necessary, for the convenience of the calculations, that the main sides intersect G at the ends of star branches.

For these reasons, the bistellar blocks whose description is detailed below are those of a fourth group, characterised in that: first, the main motifs are substantially all equal one with another and each allow the bisectrix of the corresponding main sector to be the axis of symmetry; then, the two conditions below, aim ing at simplifying industrial manufacture, are satisfied:

on the one hand, the inner contour C is made solely of portions of straight lines or of circles, whereby, like C and G, the bisectrices of the main sectors may be axes of symmetry;

on the other hand, the separatrix G is, in its final form, called form D, composed solely of straight-line segments constituting the sides of the star branches and small arcs of circle, of common radius e constituting the round-offs at the junctions of these sides.

However, it has been found that, for this shape D, the portions of flame fronts situated in the slow propergol were very close to lines made of portions of straight lines or of circles, and that conversely, if G had been traced so as to make the flame fronts in the slow propergol strictly identical to such lines, there would have been obtained a separatrix shape very close to the shape D, the sides of the branches therefore being very stretched hyperbolic arcs, and the contours of the ends of the branches becoming portions of small Descartes ovals.

It has been found then that it was possible, with a view to simplifying the calculation of the perimeters of flame fronts as from C to make a first outline of G, called shape E, defined in this way:

the sides of the branches are constituted by very stretched hyperbolic arcs, which are each constructed so as to transform the circular flame fronts of the rapid propergol into rectilinear flame fronts in the slow propergol, which meet two by two in order to create on G angular points between neighbouring branches, and which are connected by portions of small ellipses to the ends of the branches;

these ellipses confrom to an approximation of the Descartes ovals and are, to this end, constructed so as to change the flame fronts in the rapid propergol which arrive at the ends of the branches and which can be likened for this purpose to small straight-line segments, into circular flame fronts in the slow propergol;

the connection betwen an hyperbolic arc and an elliptical arc is effected either tangentially or (to simplify the calculations, in the case of a general study or of a preliminary plan) on an apex of the small axis of the el-' lipse; unless there are indications to the contrary, this is the mode of connection which will be used in the examples which follow;

finally, all the ellipses used in the construction of G are equal one to another.

However, a shape E leads to straight-line or circular flame fronts in the slow propergol only for a single value of n, namely precisely the one which has been used upon the geometrical definition of the ellipses and of the hyperbolas.

However, once the outline of the separatrix has been perfected in an E shape, it is easy to deduce from this latter a shape D which is very close thereto. There has, therefore, been obtained, with the minimum of trial and error, an outline of G which can be manufactured industrially and from which can be made without additional difficulties, if so desired, calculations of flame fronts for other values of n, more especially for values close to that retained at the start.

Finally, the shapes D and E of the separatrix can both be used in the course of the study of the blocks of the fourth group. This group is called that of the symmetrical bistellar blocks of order p and of class 0 (or more briefly blocks of order p and of class 0), the class 0 being the entire quotient (v/p), at least equal to 1.

For the shape E, each apex-image is one of the focusses, called main focus, of the ellipses forming the contour of G at the end of a branch; if this end is inside, or intersects the main side, of an elementary motif, there is only a singleelliptical arc connecting the two sides of the star branch; if the same end intersects the second side of an elementary motif, it is then formed from two small elliptical arcs symmetrical in relation to the second side of the sector and having the same main focus, situated on this second side.

By definition, a point of any closed curve surrounding 0 is called outer apex or inner apex if its distance to O is greater or smaller, respectively, than that of the points situated in its vicinity, before and after it on the curve.

For the shape D, the apex-images are substantially the centres of curvatures at the outer apices of C or C;, or of any flame front between C and C In all the cases, the separatrix G is called regular if the apex-images are all equidistant from O, and not regular in the contrary case; it is called of first sort if it has outer apices on the main sides, and of second sort in the contrary case.

The whole of the flame fronts C(t) between C,, and Cf constitutes a family of parallel curves having for orthogonal trajectories lines each formed by two segments of straight lines (or exceptionally by a single one), all leaving from C and called flame radii. In other words: ['(t) and ["(t), respectively, the portions of perimeter of C(t) situated in the rapid propergol and in the slow propergol; u '(t) and u(t), respectively, the lengths situated in the rapid propergol and in the slow propergol of a flame radius going from C to C(t). The one or the other of the lengths ['(t), 1"(1) can obviously be nil, and the same holds true for the one or the other of the lengths u(t), u(t).

The sum u(t) u(t) n u(t) is called corrected distance of the flame front C(t).

The expression corrected perimeter of C(t) is used to refer to the sum [(t) n l (t) +1"(t), where n is a coefficient chosen in such a way that [(t) represents the perimeter which C(t) ought to have, if it were situated solely in the slow propergol, so that at the instant I of the firing the pressure or the thrust obtained are substantially the same as with this flame front. Generally, n is equal to the ratio of the speeds n, and it is this value which will be adopted in the following; however, the conduct ot the calculations is absolutely similar if n a n.

It is obviously desirable that inside C the surface occupied by the propergols be the greatest possible, taking into account the space necessary for the normal flow of the burned gases through the central cavity; it is also necessary that the portion of this surface between C; and C be the smallest possible since it is the domain of the discontinuous flame fronts, having a very rapidly decreasing perimeter, therefore of poor yield (the final flame front of the firing called punctiliar flame front and marked G is reduced to a finite number of points of C,,). By definition:

the filling coefficient p is the quotient of the surface between C,, and C, divided by the inner surface of C,,; a high value of p is sought, generally at least equal to 0.9;

the theoretical residual rate is the quotient of the surface between C, and C, divided-by the surface between C and C,; a low value of ois sought, generally at the most equal to 0.05;

the shape function (1) is the quotient of the corrected perimeter of any flame front C(t) divided by the corrected perimeter of C 1) can be considered as a function of the reduced thickness y, quotient of the corrected distances of C(t) divided by that of the punctiliar flame front C generally, it is desired that the graph of d) (y) does not deviate too much (generally not by more than from a standard predetermined curve, and constituted more often than not in its major part by one or more straight-line segments which are horizontal or slightly inclined.

For all the blocks which are the objects of the invention, the graph of variation of (1) as a function of y has three successive portions, each of which has a characteristic shape.

The first is that corresponding to the rapid phase; it comprises two periods, as shown in FIG. 3 of the accompanying drawings.

The initial period (lines in dashes of the graph) concerns the flame fronts which are not made solely of arcs of circles, whose centres are on the main centres of curvature; the following are to be noted:

(1),, the value of 1) on the initial flame front, that is to say on C 11),, a first maximum, after an outline start which is generally rectilinear or polygonal.

The following period relates to flame front constituted solely by arcs of circles having centres on the main centres of curvature. The corresponding graph is a curve convex downwards; therein:

(1),. is the ordinate of the minimum (flame front C,.);

qb is the terminal ordinate, which is more often than not a second maximum (flame front C In certain cases, (1),. can merge either with (1),, or with t e It is to be noted that it is a question, in this phase as in the others, of maxima and of minima which are relative.

The second portion corresponds to the mixed phase and goes from C to C three periods are distinguished there (see FIG. 4).

The commencement period commences with the first flame front noted C which touches G, and ends with the first flame front which reaches G in the vicinities of all its inner apices; it can be reduced to the flame front C more especially when the elementary motif comprises only one inner apex of G; if it comprises more than one flame front, the inner apices contained in this motif can be touched successively by flame fronts which are then marked C C C etc. The values of :1) relative to these flame fronts are marked (b (1),, 4),,- etc., and the corresponding points of the graph are angular points, of ordinates either less than 8 or higher than (1), capable or not of being separated by points where (1) is minimum and whose ordinates are then marked (1), 1) etc.

Then comes a median period, characterised in that the number of the points of intersection of the flame fronts and of the separatrix remains equal to 2 v. The graph is then a curve convex downwards, with a minimum, of ordinate d) situated generally somewhat close to the start of the period. However, sometimes 1) is shifted to one of the ends of the period and constantly increasing or constantly decreasing.

The final period of the mixed phase commences with the first flame front, marked C,,, which reaches a point of connection on the separatrix between a side and a round-off of an outer apex. The value of 1) relative to C;,, marked 1),,, is quite often a maximum; it always presages a change in the aspect of the variation of (1). This period comprises only the flame front C,, in the extreme case where all the outer apices of G are angular and all belong to one and the same flame front which is then C,,. In the general case, it comprises other flame fronts subsequent to C,,, intersecting or touching the separatrix but capable of having with it less than 2 common points; its final flame front is C and those for which there occurs a discontinuity of the number of the points common with G are, ascending as from G, marked C C etc; on the graph, the corresponding ordinates are, in the order of the increasing y, (1) etc., b (1),, (1) these ordinates decrease rather suddenly, from (1),, to 1),,, and the representative points are angular points of the graph.

The third portion corresponds to the slow phase as shown in FIG. 5 of the accompanying drawings. The noteworthy ordinates are:

1) final minimum of the curve (y, (1)), situated generally towards the middle of the phase;

(1),, identically equal to l, for the flame front C;;

(b identically equal to 0, for the punctiliar flame front C The minimum (1) can slip either towards (1),, or towards (1 but the graph is always either rectilinear or convex downwards.

The drop of 1), from (1); l to (1),, O, has to be effected in a time which is as short as possible in .order to avoid a rupture of the chamber through excessive heating in the case of the moulded and adhered charges. It is, therefore, sought to make the difference 1y referred to as residual reduced thickness, very small, where y, is the thickness reduced relative to C,.

For example 1) arranged to be between 0.85 approximately and 1 during the entire firing.

Three geometrical magnitudes, drawn directly from C and from G, play an important part in the evolution of 1); these are:

the common value b of the distance to the centre 0 of the main centres of curvature;

the arithmetical mean a of the distances to the centre 0 of the apex-images;

the shortest distance r from the separatrix G to a main centre of curvature.

Three other geometrical magnitudes a r 6, concern the manufacture of the propergol:

ca is the greatest distance of the separatrix G to the centre 0, and is called the minimal shaping of the block; in fact, when this latter is moulded directly in a propellent by means of cores of the conventional type (that is to say non-retractable), a,,, is the minimal radius possible of the aperture of the rear bottom. For the e is the radius of curvature at the points of G which are situated in "the vicinity of an Outer apex, and where the radius'of curvature passes through a minimum; it is equal to zero if these apices are angular points; in the case of the moulded and adhered blocks, it has to be large enough to avoid cracking in the slow propergol before the moulding of the rapid propergol; as a rough assumption, it can have a value approximately equal to the value e supplied by the empirical formula:

sists, as from these data, in constructing blocks havingthe best properties. 1 r

One of the elements of the invention is a shape of the inner contour C 'considered in itself independently of any dimensional indication, called shape A, and constituted by a succession of circular arcs and of straight line portions which comprises in an elementary sector, such as that shown in FIG: 2:

a circular arc, T'T, of radius r,,, having its centre merged with a main, centre ofcurvature P situated on the main side,;inside the segment OT;

a circulanarc T11 connected tangentially to the arc TT, having its. centre at apoint M situated on the Wide of the sector and characterised by the angle OM "P=Q;

a straight line portion T 'T connected tangentially to this second circular. c and of orientation characteri sed by the angle A;

i a final circular arc T T, of radius r,,, con nected tangentially to T,T having its centre at a pointiT situated on the second side inside the segment T"M Such a shape can evolveaccording to the values of the radii of the circles andaccording to the length of its rectilinear portion. h i

Moreespecially, r r',,, )t, and the length of-the segment T T can, separately or not, be nil.

if the angle at the centre Q (equal to 1r/p), of the elementary sector is assumed known, the radius r,, of the arc of circle centred on P, and, if the evolution of the 10 function 1) in the rapid phase can be imagined, it has been found that Q" and, consequently, the position of the point M can be determined by the relations,

in which the angle Q called angle of neutrality relative to Q, is defined by the relations:

and 42 1, is the value of relating to the flame front passing through M The numerical parameters of a shape A can vary gen erally in a continuous manner within fairly large intervals, under some conditions of compatibility.

It is necessary that the line T 'I'l" T T" which has just been defined dows not leave the elementary sector. It can be seen immediately that, when the angle at P of the triangle OM "P is acute the are T T, runs the risk of intersecting the side OP if the radius r,, of the circular are T' T is less than a certain minirnu r,, function of b, 9, (2''. Likewise, the angle A 012%, where T, is the point of connection of T T, and of the segment T T must not, all things equal moreover, exceed a certain minimum.

What may be referred to as a normalised inner contour, or contour of shape N", an inner contour of shape A, is characterised in that:

r, is equal to r, when at the same time and r,, .r,; and it is equal to r,,, defined by the relation (1), in all the other cases;

. )t is equal to the angle of neutrality Q, relative to Q anddefined by the relations (4).

Experiments have shown that such contours are those allowing the greatest coefficient of filling p to be obtained for any given values of p, b, r',, and (1),, and taking 1) into account. And since the value chosen for r has generally a negligible influence on the coefficient of filling p, it is used principally to adjust to the desired value.

There has also been found a. certain number of relations to be respected between certain characteristics of the curves C and the characteristics of the curves G.

Referring now to FIG. 6 of the accompanying drawings, let an outer apex S of G situated inside, or on the sides of, an elementary sector bearing on its main side a main centre of curvature P; let P be the apex-image adjacent to S, and O the geometrical centre of the arc of circle or the ellipse belonging to G, in the vicinity of S, in the sector involved.

If G is of shape E, the distance e PP'O' is known as soon as n is assumed as well as a parameter of the arc of ellipse, for example the semi-small axis 6 and P is known to be on the segment PO.

If G is of shape D, it has been found that with an approximation sufficient in practice, and by virtue of the fact that the radius 6 of the round-off remains small, the point P can be placed on the segment PO at a distance PO from 0 such that, by laying down q P0:

The approximation made in this way has as a consequence that, if S is situated on the second side of the elementary sector, the apex-image relative to this apex is likened to two separate points according to whether the one or the other of the elementary sectors is considered as having this second side; however, this is not a drawback in practice, and moreover the two points in question are at the same distance from the centre 0.

It has been found, on the one hand that, in order to minimise o", it is desirable to place the outer apices of G on radii of the circle C,. cutting-up on this circle 11 'sectors of angles at the apices all substantially equal to then a and b are given exactly or with an approximation sufficient in practice by the relations below, where w, is called angle of neutrality relative to to:

On the other hand, P,- and O, are used to designate (i 1, 2, 3 j) the j apex-images and the j geometrical centres of the ends of a separatrix, of shape D or, which are situated inside or on the sides of one and the same elementary sector; 6,- refers to the distances P,- O,-, q, the distances PP,- of the apex-images at the main centre of curvature P of the sector, and a; the distances of the same apex-images from the centre 0; and it has been found:

that it is advantageous for the convenience of calculation to place, on j radii of C whose angles with a main side of the sector are worth either (2J1) w or 2] to, with J 1, 2, etc., j, the geometrical centres 0', when the separatrix is of shape D, or the apex-images P,- when it is of shape E;

that in the two cases, the j lengths q,- PP,- are the roots of a system of j equations expressing, on the one hand, the fact that a is the arithmetical mean of the v distances to the centre 0 of the apex-images relating to the v apices of the separatrix, on the other hand, the equality of value of the j different expressions which are obtained for the corrected distance of C when each of the j flame radii passing through the apices-images P,- is followed, this equality being able to be expressed 12 by the j-l equations below: (9) U =U U,-, with U,=na,- (n-l) ',--q,-. and i= 1, 2-. etc.,j.

It will be apparent that the formulae given above and allowing the determination of the positions of the outer apices of the separatrix G are not absolutely mandatory, since they may have a certain number of reasonable approximations. Likewise, upon the effective realisation of the blocks of propergols in accordance with the invention, the theoretical positions of the outer apices of G, determined by the said mathematical formulae, will be able to be more or less respected as a result of the imperfections inherent in the physical processes used. It follows that blocks made in accordance with the invention cannot be limited to those for which the outer apices of G obey strictly the ideal positioning defined by the said formulae, but also extend to the other blocks of the same type in which the positionings of the outer apices of G are close to these theoretical positions.

Let R,-' (1'' l, 2 .j) be the inner apices ofG situated inside or on the sides of the elementary sector and R,- I the points of G situated in the vicinity of the R, and such that their distances to the main centre of curvature P of the same sector pass through a relative minimum (the R*,- can be wholly or partly merged with the R,- I To detennine the R*,- I comes back to determining the R,- v The term r, denotes the length of the segment PR, 6 the acute angle of this segment and of the main side, and r the smallest of the lengths r r, is equal to the radius of the arcs of circle constituting C and it has been found that its expression as a function of (b is given by the equations:

first of all, they have to be spaced out so that d) evolves, just after the flame front C in the sense desired; generally, it is desired that at the start of the mixed phase dz no longer increases, or at least begins very soon to decrease;

then, they must not be too small, so that the branches of G have a length sufficient to have a correct action on the flame fronts coming from the rapid propergol; if two sides of G having in common an inner apex R were too short, (1 could rise too much before the flame fronts reach R; then, this point R being reached, (1) would drop in a very pronounced manner, arriving at the end of the mixed phase at a value lower than that which would have been given by longer branches;

finally, they can satisfy various desirable convenient factors; for example, if they are all taken to be equal, the calculations of d; in, .the mixed phase will be facilitated; or, for the sake of regularity, they can be taken such that the distances to the centre 0 of all the inner apices are equal.

As for the angles O I they are determined so as to satisfy the imperatives of variation of (b and to facilitate calculation; they are chosen, preferably, such that the inner apices of G are not too remote from the bisectrices of the angles formed by the radii of C passing through two consecutive outer apices of G.

All the determinations below are made after the values of p, c, n and noteworthy values of d) have been as- 13 sumed. These latter are supplied as desired by the us ers; p and c are chosen mainly as a function of the de sired values of p and of with the assistance of the tables of results of Example 1 below. As for n, it is determined by trial and error, by trying to construct the block with a certain value and by observing if the curve of evolution of (b which results therefrom is acceptable; a complete illustration of the method and of the influence of n on the function of shape is given in Example 3 below. However, it can be said in general that, if the usual conditions of maximum or of minimum are imposed on the variations of 15, there exists, for a wide range of values of p and of c, a certain interval of the possible values of n; it has been found that the mean point of this interval varies grosso modo in the converse sense of v, and is slightly less than 2 for v 24.

At a point situated on the axis of the block and such that the plane 11' normal to the axis at this point intersects the lateral surface of the central cavity, the inherent locking (serrage propre) is by definition the quotient, of the area of the portion of surface of the central cavity which generates, at the start of the firing, gases directed towards the plane 72' divided by the area of the section of the central cavity according to 71'.

On the whole of the points above, the inherent locking (serrage propre) has a maximum K which is a characteristic of the block. Now, it is necessary that K does not exceed a certain limit, as a function of the propergols used; and this circumstance is often troublesome, with blocks of somewhat elongated shape, when one uses fully the possibilities which are given by the invention of reducing the surface of the inner contour and of increasing, accordingly, the coefficient of filling.

A substantial improvement in this field is supplied by a very general family of bistellar blocks, characterised in that, if the sections of the axis cavity be considered to be made successively, through planes normal to the axis, always following the same flow direction of the gases, the first of these sections, situated at the origin of the flow, has a surface less than that of the final section, situated at the outlet of the block, and the surfaces of the intermediate sections never decrease; there can thus be obtained for the same value of K a central cavity of volume less than that of which the sections would all have the same surface.

More especially, the invention allows the manufacture of bisteller blocks called of the fifth group, belonging to the general family which has just been de scribed, and characterised moreover in that:

they form part of the fourth group, that is to say, of that of the blocks of order p and of class c;

the sections of their cylindrical portion bear outlines of the flame front C all substantially identical;

the sections, through planes normal to the axis of the lateral surface of their central cavity, are all contours C ibelonging to the form A described above; for a given block, these contours are each constructed with the same values of Q, 1), r r' if these are considered one after the other by always following the same direction of flow of the gases, a finite number m of different contours are found, which are distinguished one from the other only by the values of hand, accessorily, of (2.

Thus, the central cavity is composed of m successive portions, each containing sections having identical contours, and occupying on the axis of the block successive lengths marked Z Z2 z, in the direction of the flow of the gases.

The length of the portion, contained in an elementary sector, of an inner contour of shape A is marked s and called elementary inner perimeter. On the m successive contours C of a block of the fifth group, considered in the same order as above, there is noted s s,,, the elementary perimeters, p p p the coefficients of filling, and A A A, the values of A; the co efficient of filling p and the elementary inner perimeter s of the block are by definition the quotients, of the sums p z p 1 p z and s z S2Z1+. s,,,z,,, divided by the sum z Z2 z,,,.

In order to determine a block of the fifth group, one can start from the outline, called reference outline", of an inner contour C of shape A meeting the conditions imposed, corresponding, to values marked Q*, Q"*, M, b, r,,, r,,* of the parameters of definition, and characterised itself by a coefficient of filling p* and by an elementary inner perimeter 3*.

If a block of given length and with a central cavity were made whose sections normal to the axis are all identical to the reference outline, there would be obtained for the maximum inherent locking (serrage propre) a value K which would by hypothesis be too great; the problem is therefore to define a block of the fifth group, of same outer dimensions, which is constructed with values of O, b, r r identical to those of the reference outline, which retains the values p* and s* of p and of s, and for which the maximal inherent locking (serrage propre) has a value less than K* and as low as possible.

It has been found, in practising the invention, that the block sought is theoretically characterised in that:

m has to be of the largest possible value and s,,, of the smallest possible value;

since there exists, with the conditions laid down, a linear relationship between the p and the s,- (i l, 2 m) which can be written s s; B( 1*91), 5,, and B being positive constants, the elementary inner perimeters s,- have to form, with the constant s,,, a decreasing geometrical progression of m l successive terms s the z,- (i= 1, 2 m) have to all be equal one to another.

However, in practice:

m cannot be very large, without excessive complications in practice; and, moreover, its influence on the value of K is rather low;

s,,, cannot be less than the value of s for which A is nil;

5 cannot exceed a certain maximum corresponding to a line T T T T T of FIG. 2 which would be, of course, situated inside the elementary motif and of which, furthermore, the shortest distance to the main side of the same sector would have a minimum value not nil 2;, still compatible with a good circulation of the combustion gases along the inner surface of the central cavity.

It has been found that, for m fixed and for s, chosen the greatest possible, the optimal block is that for which the numbers s s s, form a geometrical progression and the numbers 2 Z12 z are all equal. It is therefore sufficient to settle: s as a function of the value tolerated for 5; there can be deduced therefrom by calculation, since s* and m are known, the terms of the sequence s s s,,,.

However, it can happen that the final term or some of the final terms of this sequence are less than the value of s corresponding to it 0. It has been observed in this case that without notice able disadvantage, for the cor- 

1. A solid propellant for use as a gas generator, particularly for the reaction propulsion of vehicles in space or in a gaseous or liquid medium, said propellant having a lateral outer surface which is a cyclinder of revolution and which does not participate in combustion and the propellant having a lateral inner surface defining a central cavity of a shape elongated in the direction of the axis of the cylinder, the inner surface being ignitable at the initial instant of firing, and wherein said solid propellant is formed from two propergols having different speeds of combustion, the two propergols contacting one another without interruption along a continuous separation surface or separatrix surrounding the inner surface, the more rapidly combustible propergol occupying the space between the inner surface and the separatrix and the more slowly combustible propergol occupying the remaining volume of the solid propellant between the separatrix and the outer surface and wherein the solid propellant through a plane perpendicular to the axis of the cylinder satisfies the following two conditions:
 1. the inner surface is closed, continuous, and is substantially star-shaped having p branches with p at least equal to 3, the shape of the star being such that the rapid phase of the combustion in which only the rapidly combustible propergol burns, ends at the instant when the section of the surface of combustion is a line formed only of p consecutive arcs having their concavity towards a point and forming a curve parallel to a portion of the end of a different branch of the star, said arcs being composed substantially of circular arcs whose centers of curvature are situated on p radii of said inner surface, each radius intersection said inner contour at a point of the end of a different star branch; and
 2. the separatrix does not touch the outer surface or the inner surface and has substantially the shape of a star having Nu branches, with Nu at least equal to p, the separatrix star shape being such that toward the end of the slow phase of the combustion in which only the more slowly combustible propergol burns, the cross section of the surface of combustion which reaches said outer surface is a line formed solely of Nu consecutive arcs having their concavity towards said point, being substantially tangential to said outer surface, and being placed opposite the end of one of said Nu branches of said separatrix, said Nu arcs being themselves substantially composed of circular arcs whose centers are all situated in the vicinities of the ends of said Nu star branches.
 2. A solid propellant as claimed in claim 1 wherein the inner sUrface at the end opposite that from which the gases formed upon combustion exit has a surface less than the surface at the end from which the gases exit and the surface increases from the opposite end to the end from which the gases exit.
 2. the separatrix does not touch the outer surface or the inner surface and has substantially the shape of a star having Nu branches, with Nu at least equal to p, the separatrix star shape being such that toward the end of the slow phase of the combustion in which only the more slowly combustible propergol burns, the cross section of the surface of combustion which reaches said outer surface is a line formed solely of Nu consecutive arcs having their concavity towards said point, being substantially tangential to said outer surface, and being placed opposite the end of one of said Nu branches of said separatrix, said Nu arcs being themselves substantially composed of circular arcs whose centers are all situated in the vicinities of the ends of said Nu star branches.
 3. A solid propellant as claimed in claim 1, wherein said outer surface has p'' circular sectors, with p'' at least equal to p, so that the portions of the surface of combustion contained in said sectors are all superimposable one on the other and wherein all the portions of said separatrix contained in the same sectors are similarly superimposable.
 4. A solid propellant as claimed in claim 1, wherein the portions of the combustion surface and the separatrix contained in any main sector of said outer surface limited by two consecutive main sides each has, as an axis of symmetry, the inner bisectrix of the said main sector; and wherein the sides of said main sectors each intersect the separatrix at a point of the end of one of the Nu star branches.
 5. A solid propellant as claimed in claim 3, wherein the portions of said surface of combustion and said separatrix contained in any main sector each have the inner bisectrix of said sector as an axis of symmetry.
 6. A solid propellant as claimed in claim 5, wherein the portion of said inner surface situated in any main sector and subtending an angle at the center of 2 Omega 2 pi /p, comprises substantially the successive elements, connecting tangentially one to the other in order from a side to the bisector of 2 Omega a first circular arc having a radius ro, centered on the side of the sector of curvature P, which is on said side of said sector, inside at a distance b from the axis a second circular arc having its center at a point situated on the bisectrix of the sector and characterized by the angle Omega ''''; from the point on the bisector to the axis and from the point on the bisector to the main center of the curvature a straight line section of length zeta , and of orientation characterized by the angle lambda from the point on the bisector to the axis and from the point on the bisector to the beginning of the straight line section and a third circular arc having its end on the bisector between the axis and the point on the bisector and having its center at a point on the segment from the end of sector to the point on the bisector.
 7. A solid propellant as claimed in claim 6, wherein the following values substantially apply: ro equal to
 8. A solid propellant as claimed in claim 6, wherein that at least one of the values ro, r''o, zeta , lambda is substantially nil.
 9. A solid propellant as claimed in claim 6 wherein
 10. A solid propellant as claimed in claim 7, wherein at least one of the values ro, r''o , zeta , lambda is substantially nil.
 11. A solid propellant as claimed in claim 1, wherein the sides of the Nu star branches are comprised by portions of straight lines or hyperbolas and the ends of the branches are portions of a circle, a Descartes oval, or an ellipse.
 12. A solid propellant for reaction propulsion having a substantially cylindrical outer surface and which comprises two separate substantially homogeneous propergols, the first of which has a combustion speed which is higher than the combustion speed of the second propergol, the first propergol defining a central axial cavity in the cylinder and extending inwardly from said cavity to the second propergol and the second propergol occupying the volume between the first propergol and the cylindrical outer surface, the cross-section of the surface of the first propergol at the interface with the cavity perpendicular to the axis of the cylinder being in substantially the shape of a star with p branches, p, being at least three, the outer surface of the first propergol defining a separatrix which is coextensive with the inner surface of the second propergol the cross-section of the surface of the separatrix perpendicular to the axis of the cylinder having the form of a star having Nu branches wherein Nu is greater than p, the maximum distance between the separatrix and the axis of the cylinder being not more than about 0.75 times the radius of the cylinder.
 13. A reaction propulsion device comprising a propellant according to claim 12 capable of producing gases upon combustion and a generally cylindrical chamber enclosing the propellant, the chamber having at least one aperture to permit the efflux of gases from the chamber. 